Table Of ContentExploring
Mathematical
Modeling in Biology
Through Case Studies
and Experimental
Activities
Exploring
Mathematical
Modeling in Biology
Through Case Studies
and Experimental
Activities
Rebecca Sanft
University of North Carolina Asheville
Asheville, NC, United States
Anne Walter
St. Olaf College
Northfield, MN, United States
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Contents
Preface ...................................................... ix
Tipsforusingthistext–instructors ................... x
Tipsforusingthistext–students ..................... xi
Onlineresources ................................. xii
Acknowledgments ............................................. xiii
UNIT 1 Preliminaries: models, R, and lab techniques ......... 1
1.1 Bringingmathematicsandbiologytogetherthroughmodeling .... 1
1.2 Rbasics ............................................. 4
1.2.1 RStudiolayout .................................. 5
1.2.2 Simplecalculations ............................... 6
1.2.3 Datastructures .................................. 7
1.2.4 Basicplotting ................................... 11
1.2.5 Readingdatafromfiles ............................ 14
1.2.6 Iteration ....................................... 15
1.2.7 Fittingalinearregressionmodel ..................... 17
1.3 Prelablab:practicingthefundamentals ..................... 24
UNIT 2 Introduction to modeling using difference equations ... 29
2.1 Discrete-timemodels ................................... 30
2.1.1 Solutionstofirst-orderdifferenceequations ............. 32
2.1.2 Usinglinearregressiontoestimateparameters ........... 34
2.2 Puttingitalltogether:thewhoopingcrane ................... 38
2.3 Casestudy1:Islandbiogeography ......................... 42
2.3.1 Background..................................... 42
2.3.2 Modelformulation ............................... 43
2.3.3 Rakatastory .................................... 45
2.3.4 Modernapproach:lineagedata ...................... 48
2.3.5 BacktoMacArthurandWilson:effectsofdistanceandarea 51
2.4 Casestudy2:Pharmacokineticsmodel ...................... 54
2.4.1 Background..................................... 54
2.4.2 Formulatingthemodel............................. 57
2.4.3 Understandingthemodel ........................... 58
2.4.4 Parameterestimation .............................. 60
2.4.5 Modelevaluation/analysis .......................... 63
2.4.6 Furtherexploration ............................... 63
2.5 Casestudy3:Invasiveplantspecies ........................ 66
2.5.1 Background..................................... 66
2.5.2 Modelformulation ............................... 67
2.5.3 Parameterestimation .............................. 69
v
vi Contents
2.5.4 Modelpredictions ................................ 71
2.5.5 Managementstrategies ............................ 73
2.6 Wetlab:logisticgrowthmodelofbacterialpopulationdynamics .. 77
2.6.1 Introduction .................................... 77
2.6.2 Modelingpopulations ............................. 78
2.6.3 Theexperiment .................................. 80
2.6.4 Modelcalibrationandanalysis....................... 84
2.6.5 Experimentpart2:effectofchangingmedia ............ 85
UNIT 3 Differential equations: model formulation, nonlinear
regression, and model selection ..................... 87
3.1 Biologicalbackground .................................. 87
3.2 MathematicalandRbackground .......................... 89
3.2.1 Differentialequation-basedmodelformulation .......... 89
3.2.2 Solutionstoordinarydifferentialequations ............. 96
3.2.3 Investigatingparameterspace ....................... 97
3.2.4 Nonlinearfitting ................................. 98
3.3 Modelselection ....................................... 106
3.4 Casestudy1:Howleafdecompositionratesvarywith
anthropogenicnitrogendeposition ......................... 108
3.4.1 Background..................................... 108
3.4.2 Thedata ....................................... 109
3.4.3 Modelformulation ............................... 109
3.4.4 Parameterestimation .............................. 111
3.4.5 Modelevaluation ................................. 112
3.5 Casestudy2:Exploringmodelstodescribetumorgrowthrates ... 113
3.5.1 Background..................................... 113
3.5.2 Thedata ....................................... 113
3.5.3 Modelformulation ............................... 114
3.5.4 Parameterestimation .............................. 115
3.5.5 Modelevaluation:descriptivepower .................. 117
3.5.6 Modelevaluation:predictivepower ................... 118
3.6 Casestudy3:Predatorresponsestopreydensityvarywith
temperature .......................................... 120
3.6.1 Background..................................... 120
3.6.2 Analysisoffunctionalresponsedata:determiningthe
parameters...................................... 123
3.6.3 Exploringfunctionalresponsesasafunctionoftemperature. 127
3.7 Wetlab:enzymekineticsofcatecholoxidase ................. 130
3.7.1 Overviewofactivities ............................. 130
3.7.2 Introductiontoenzymecatalyzedreactionkinetics ........ 130
3.7.3 Derivingthemodel ............................... 133
3.7.4 EstimatingK andV ........................... 135
M max
3.7.5 Ourenzyme:catecholoxidase ....................... 137
Contents vii
3.7.6 Experiment:collectinginitialratesfortheMichaelis–Menten
model ......................................... 139
3.7.7 Effectsofinhibitorsonenzymekinetics ................ 142
3.7.8 Experiment:measuringtheeffectsoftwocatecholoxidase
inhibitors,phenylthioureaandbenzoicacid ............. 144
UNIT 4 Differential equations: numerical solutions, model
calibration, and sensitivity analysis .................. 147
4.1 Biologicalbackground .................................. 147
4.2 MathematicalandRbackground .......................... 149
4.2.1 Numericalsolutionstodifferentialequations ............ 149
4.2.2 Calibration:fittingmodelstodata .................... 154
4.2.3 Sensitivityanalysis ............................... 157
4.2.4 Puttingitalltogether:thedynamicsofEbolavirusinfecting
cells .......................................... 167
4.3 Casestudy1:Modelingthe2009influenzapandemic ........... 175
4.3.1 Background..................................... 175
4.3.2 TheSIRmodel .................................. 176
4.3.3 Cumulativenumberofcases ........................ 179
4.3.4 Epidemicthreshold ............................... 180
4.3.5 Publichealthinterventions.......................... 182
4.3.6 2009H1N1influenzapandemic ...................... 183
4.4 Casestudy2:Optimizingimmunotherapyinprostatecancer...... 187
4.4.1 Background..................................... 187
4.4.2 Modelformulation ............................... 189
4.4.3 Modelimplementation............................. 193
4.4.4 Parameterestimation .............................. 193
4.4.5 Vaccinationprotocolsandmodelpredictions ............ 196
4.4.6 Sensitivityanalysis ............................... 197
4.4.7 Simulatingothertreatmentstrategies .................. 198
4.5 Casestudy3:Quorumsensing ............................ 199
4.5.1 Introduction .................................... 199
4.5.2 Modelformulation ............................... 201
4.5.3 Parameterestimation .............................. 203
4.5.4 Modelsimulations ................................ 204
4.5.5 Sensitivityanalysis ............................... 207
4.6 Wetlab:hormonesandhomeostasis—keepingbloodglucose
concentrationsstable ................................... 211
4.6.1 Overviewofactivities ............................. 211
4.6.2 Introductiontobloodglucoseregulationanditsimportance . 211
4.6.3 Developingamodel............................... 213
4.6.4 Experiment:measuringbloodglucoseconcentrations
followingglucoseingestion ......................... 218
4.6.5 Analysis ....................................... 220
viii Contents
4.6.6 Thoughtstoconsiderforpotentialfollow-upexperiments .. 222
UNIT 5 Technical notes for laboratory activities .............. 225
5.1 Introduction .......................................... 225
5.2 Populationgrowth ..................................... 225
5.3 Enzymekinetics....................................... 229
5.3.1 Notesonotherenzymesorsimilarexperiments .......... 231
5.4 Bloodglucosemonitoring ............................... 232
5.4.1 Tipsforglucosemonitoring ......................... 232
5.4.2 Otherlabactivities ............................... 234
Bibliography.................................................. 237
Index ....................................................... 241
Preface
This mathematics of biology text is a stand-alone compendium of exercises, cases,
andwetlabsdesignedtohelpmathematicsandlifesciencescollegestudentsintegrate
mathematical,computational,andresearchapproachestoaddressingrealproblems.
Thesematerialsaredesignedforanycourseinthebroadareaofmathematicalbiology
forundergraduates,particularlyfillinganicheforclassesintendedforstudentsfrom
multipledisciplines.Thereisanongoingparadigmshiftinhowscienceispracticed
and taught, from a strong disciplinary focus to one of interdisciplinary approaches
to address a common problem. One critical interface is that between mathematics
and biology, which is exploding with possibilities due to the emergence of power-
fulcomputationalanddatacollectiontoolsallowingmodelsanddatatoconvergein
hypothesistestingandevaluation.
Thiscompendiumprovidesasetofmaterialstoactivelyengagestudentsinmath-
ematical modeling of biological phenomena while simultaneously interacting with
realdatagatheredfromtheliteratureorexperimentallybythem.Weintendstudents
to experience how the act of modeling brings clarity to complex systems and often
identifies or helps scientists extract parameters that are most critical or unavailable
experimentally. Students will see the utility of models for identifying further ques-
tions,predictingoutcomes(drugdosingplans,invasivespeciespopulationgrowth,or
effects of global warming) or understanding complex systems (community popula-
tiondynamics,feedbackloops,andhormonalcontrol).
Thistextistrulyinterdisciplinary.Itprovidescasestudiesacrossmultiplelevels
of organizationand areas of biologicalinquiry and includes instructions for several
wetlabssothatstudentscancollecttheirowndatatouseinmodelformulationand
validation.Eachunithasspecificlearninggoals,mathematicalbackground,embed-
ded R code, and examples and exercises in the form of case studies and a lab, thus
making it easy to adapt some parts or all of this text for a wide range of courses
that integrate mathematics and biology from the level of several homework assign-
mentsoracasestudyuptoanentirecourse.Anexampleofthelatterbasedonour
experienceisgivenin[61].
The audience for this compendium of lessons, exercises, case studies, and lab
exercisesisprimarilyundergraduatelifesciencesormathematicsstudentsandtheir
faculty.Itisdesignedasafirstcourseinmathematicalmodelingwithafocusonbio-
logicalproblems.Althoughthetargetaudienceissophomores,wehavehadstudents
atallundergraduatelevelssuccessfullylearnfromandenjoythesematerials.Wedo
assumethereaderhashadsingle-variablecalculusbutnocollege-levelbiology;the
labsaretypicalofthoseinnonmajorscourseswithsignificantlyhigherexpectations
for data analysis. Due to the latter requirement, advanced biology students benefit
tremendouslyfromlookingatanexperimentfromadifferentpointofview.
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